{
  "id": "1612.00597",
  "version": "v1",
  "published": "2016-12-02T08:57:44.000Z",
  "updated": "2016-12-02T08:57:44.000Z",
  "title": "On the Bohr inequality",
  "authors": [
    "Yusuf Abu Muhanna",
    "Rosihan M. Ali",
    "Saminathan Ponnusamy"
  ],
  "journal": "29 pages; Will be published by \"Springer International Publishing AG 2016\", N.K. Govil et al. (eds.), Progress in Approximation Theory and Applicable Complex Analysis, Springer Optimization and Its Applications 117",
  "doi": "10.1007/978-3-319-49242-1_13",
  "categories": [
    "math.CV"
  ],
  "abstract": "The Bohr inequality, first introduced by Harald Bohr in 1914, deals with finding the largest radius $r$, $0<r<1$, such that $\\sum_{n=0}^\\infty |a_n|r^n \\leq 1$ holds whenever $|\\sum_{n=0}^\\infty a_nz^n|\\leq 1$ in the unit disk $\\mathbb{D}$ of the complex plane. The exact value of this largest radius, known as the \\emph{Bohr radius}, has been established to be $1/3.$ This paper surveys recent advances and generalizations on the Bohr inequality. It discusses the Bohr radius for certain power series in $\\mathbb{D},$ as well as for analytic functions from $\\mathbb{D}$ into particular domains. These domains include the punctured unit disk, the exterior of the closed unit disk, and concave wedge-domains. The analogous Bohr radius is also studied for harmonic and starlike logharmonic mappings in $\\mathbb{D}.$ The Bohr phenomenon which is described in terms of the Euclidean distance is further investigated using the spherical chordal metric and the hyperbolic metric. The exposition concludes with a discussion on the $n$-dimensional Bohr radius.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-12-02T08:57:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30A10",
      "30B10",
      "30B50",
      "30C35",
      "30C45",
      "30C80",
      "30H05",
      "32A05",
      "32A07",
      "32A10",
      "46L06",
      "47A56",
      "47A13"
    ],
    "keywords": [
      "bohr inequality",
      "largest radius",
      "dimensional bohr radius",
      "concave wedge-domains",
      "punctured unit disk"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}